On 07/02/2020 04:06, Keith F. Lynch wrote:
Most such values of x are transcendental numbers. But not all. What rational values can x have? Can it have an irrational algebraic value?
(This is values of x where m^x - 2n^x = 1, with m,n positive integers.) I betcha x is never an irrational algebraic number, and I betcha there will not in our lifetimes be sufficient mathematical technology to prove this.
Hence my second table. I assume they're all transcendental except of course where x=y, in which case z=x=y. Can anyone prove this? Or find a counterexample?
(This is values of x where m^x + n^x = 2x^x, with m,n positive integers.) I betcha x is never algebraic unless m=n, and I betcha there will not in our lifetimes be sufficient mathematical technology to prove this. (I could be wrong about either; I'm not a transcendence expert; but this sort of thing is usually Really Hard.) -- g